- The paper establishes that bottom-up holographic models with candidate DSI retain full AdS conformal symmetry through analytical Killing vector analysis.
- It demonstrates that periodic warp factors and log-periodic corrections arise without violating energy conditions or destabilizing the bulk.
- The work constrains the feasibility of engineering genuine scale invariance without full conformal invariance and informs the holographic swampland program.
Discrete Scale Invariance in Holography: Symmetry, Constraints, and Holographic Implications
Introduction
This work scrutinizes the embedding of discrete scale invariance (DSI) within the holographic AdS/CFT paradigm, focusing on a specific 5+1-dimensional bottom-up gravitational model pioneered by Balasubramanian. The intent is to analyze the precise symmetry structure of the corresponding bulk solutions and assess their physical and holographic consistency. The broader context involves characterizing under what circumstances DSI—manifest in cyclic RG flows as opposed to ordinary fixed points—can be realized and understood within a consistent holographic framework.
Discrete Scale Invariance and Holographic Model Building
The distinction between continuous and discrete scale invariance is at the core. While ordinary scale invariance is symmetry under arbitrary rescaling of spacetime coordinates, DSI restricts this to a discrete set of scale factors. Phenomenologically, DSI is associated with log-periodic corrections and complex critical exponents, observed across a variety of physical systems (from fractals to Efimov states and critical gravitational collapse).
In the context of AdS/CFT, standard wisdom suggests that bulk geometries with explicit discrete scale invariance are difficult to realize without violating energy conditions or sacrificing stability. The holographic c-theorem places stringent constraints, typically forbidding periodic dependence in warp factors unless the null energy condition (NEC) is violated. Balasubramanian's construction circumvents this by engineering bulk metrics with warped extra dimensions and axion-like scalar fields, yielding solutions where the NEC is satisfied, and the metric acquires periodic dependence through Jacobi elliptic functions.
A central claim of the referenced paper is the demonstration that the full conformal isometry group of AdS is retained in a non-trivial manner in the candidate DSI bulk geometries. This is established by solving the Killing equations analytically and showing the existence of 15 independent Killing vector fields. This algebra, isomorphic to so(4,2), signals that the underlying spacetime actually possesses full AdS symmetry, even though the DSI ansatz naively suggests spontaneous symmetry breaking down to Poincaré plus discrete scale transformations.
This result has significant theoretical implications:
- The metric does not genuinely break conformal invariance in the bulk, despite the log-periodic structure of invariants (e.g., the Ricci scalar).
- The DSI and cyclic RG flow behavior anticipated on the boundary are, in fact, holographically dual to a (possibly field-redefined) conformal fixed point, consistent with expectations from field theory analyses of cyclic RG and limit cycle flows [e.g. the field-theoretical discussions in (Luty et al., 2012, Fortin et al., 2012)].
- The model exhibits a vanishing Weyl tensor for all parameter values, reinforcing the maximal symmetry.
This analytic killing vector approach not only resolves the ambiguity regarding the symmetry content of these models but sets a benchmark for evaluating DSI in future proposed holographic models.
The deeper question addressed is whether one can engineer a holographic bulk theory that is scale invariant (or discretely scale invariant) but not conformally invariant at the level of the isometry algebra. The analytic results indicate that, at least for the specific bottom-up ansatz studied, the presence of a D-type (dilatation) Killing vector in tandem with the Poincaré algebra generically implies the existence of the full conformal algebra due to the structure of the Killing equations. Attempts to realize scale invariance without conformal invariance in a physically consistent bulk remain frustrated by overdetermined constraints.
Existing literature (e.g., Nakayama's reviews on scale vs. conformal invariance in holography (Nakayama, 2013)) also buttresses the conclusion that, under reasonable energy and regularity conditions, scale invariance in the bulk is essentially equivalent to conformal invariance. The current work makes this equivalence explicit for the considered models and clarifies parameter regimes where the situation might differ.
Holographic Consistency and Landscape Considerations
The discovery that ostensibly DSI bulk geometries retain hidden full conformal symmetry prompts new scrutiny of the so-called "swampland" of holography. Specifically, it sharpens the logic concerning which bulk geometries admit consistent, unitary field theory duals. Four logical possibilities are advanced:
- DSI is prohibited in both bulk and boundary by deep theorems (e.g., the a/c-theorems or constraints derived via energy conditions).
- DSI bulk models exist but belong to the "forbidden landscape" without proper duals—a phenomenon observed in other contexts in the swampland literature.
- DSI is allowed in both bulk and boundary, permitting robust AdS/CFT constructions for cyclic RG flows.
- Boundary DSI is possible but has no gravitational dual—using this as a probe of holographic correspondence limits.
The analysis demonstrates that in the studied bottom-up model, scenario (1) appears most consistent: efforts to realize bulk DSI geometry without underlying conformal symmetry automatically rebuild the full symmetry algebra.
Prospects for Physical Realization and Further Directions
The work remarks on the relation between discrete scale invariance in holography and more general contexts such as MERA tensor networks, fractal geometries, and systems engineered on fractal lattices. The possible presence of DSI at the lattice or network level that does not survive as a new symmetry in the continuum (holographic) limit is an interesting direction for both theory and experiment. The role of stability, particularly in non-supersymmetric axionic compactifications, remains open. The top-down model surveyed in (Balasubramanian, 2013), which does not exhibit hidden conformal isometries for generic parameter choices, is highlighted as a potential avenue for isolating bulk models where true DSI without full conformal symmetry might manifest, subject to stability analysis and holographic dictionary matching.
Conclusion
This study provides a sophisticated analysis of the symmetry content of bottom-up holographic models with candidate discrete scale invariance. By exhaustively classifying Killing vector fields, the work demonstrates that the putative breaking of AdS conformal symmetry to discrete scale invariance is illusory in these models: the full isometry group survives in a non-trivial, "hidden" fashion. The implications are manifold for the structure of holographic dualities, the swampland program, and the search for non-trivial scale-invariant but non-conformal RG dynamics with gravitational duals. Progress in constructing explicit physically stable holographic DSI solutions, especially in top-down or string-theoretic contexts, remains contingent on further developments in both field theory and gravitational model building.
Reference:
"Discrete scale invariance in holography revisited" (1711.03113)